= Solution
The quotient is defined only when $z+\overline u\ne0$. Its unit-modulus condition then gives $|z+u|^2=|z+\overline u|^2$. On expanding with <complex conjugates>,
$$
\begin{aligned}
0&=(z+u)(\overline z+\overline u)-(z+\overline u)(\overline z+u)\\
&=(z-\overline z)(\overline u-u)
=4\operatorname{Im}z\,\operatorname{Im}u.
\end{aligned}
$$
Hence at least one imaginary part is zero, and \b[either $z$ or $u$ is real]. Conversely, either reality condition makes the two squared moduli equal; provided the denominator is nonzero, the quotient has modulus one. No conclusion is drawn from an undefined quotient.
Back to article page
